3.708 \(\int \frac{1}{x^5 \sqrt [3]{a+b x^2}} \, dx\)

Optimal. Leaf size=138 \[ \frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{6 a^{7/3}}+\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3}}-\frac{b^2 \log (x)}{9 a^{7/3}}+\frac{b \left (a+b x^2\right )^{2/3}}{3 a^2 x^2}-\frac{\left (a+b x^2\right )^{2/3}}{4 a x^4} \]

[Out]

-(a + b*x^2)^(2/3)/(4*a*x^4) + (b*(a + b*x^2)^(2/3))/(3*a^2*x^2) + (b^2*ArcTan[(
a^(1/3) + 2*(a + b*x^2)^(1/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)) - (b^2*Lo
g[x])/(9*a^(7/3)) + (b^2*Log[a^(1/3) - (a + b*x^2)^(1/3)])/(6*a^(7/3))

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Rubi [A]  time = 0.228244, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{6 a^{7/3}}+\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3}}-\frac{b^2 \log (x)}{9 a^{7/3}}+\frac{b \left (a+b x^2\right )^{2/3}}{3 a^2 x^2}-\frac{\left (a+b x^2\right )^{2/3}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a + b*x^2)^(1/3)),x]

[Out]

-(a + b*x^2)^(2/3)/(4*a*x^4) + (b*(a + b*x^2)^(2/3))/(3*a^2*x^2) + (b^2*ArcTan[(
a^(1/3) + 2*(a + b*x^2)^(1/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)) - (b^2*Lo
g[x])/(9*a^(7/3)) + (b^2*Log[a^(1/3) - (a + b*x^2)^(1/3)])/(6*a^(7/3))

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Rubi in Sympy [A]  time = 15.961, size = 126, normalized size = 0.91 \[ - \frac{\left (a + b x^{2}\right )^{\frac{2}{3}}}{4 a x^{4}} + \frac{b \left (a + b x^{2}\right )^{\frac{2}{3}}}{3 a^{2} x^{2}} - \frac{b^{2} \log{\left (x^{2} \right )}}{18 a^{\frac{7}{3}}} + \frac{b^{2} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}} \right )}}{6 a^{\frac{7}{3}}} + \frac{\sqrt{3} b^{2} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{2}}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b*x**2+a)**(1/3),x)

[Out]

-(a + b*x**2)**(2/3)/(4*a*x**4) + b*(a + b*x**2)**(2/3)/(3*a**2*x**2) - b**2*log
(x**2)/(18*a**(7/3)) + b**2*log(a**(1/3) - (a + b*x**2)**(1/3))/(6*a**(7/3)) + s
qrt(3)*b**2*atan(sqrt(3)*(a**(1/3)/3 + 2*(a + b*x**2)**(1/3)/3)/a**(1/3))/(9*a**
(7/3))

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Mathematica [C]  time = 0.0539015, size = 82, normalized size = 0.59 \[ \frac{-3 a^2-4 b^2 x^4 \sqrt [3]{\frac{a}{b x^2}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{a}{b x^2}\right )+a b x^2+4 b^2 x^4}{12 a^2 x^4 \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a + b*x^2)^(1/3)),x]

[Out]

(-3*a^2 + a*b*x^2 + 4*b^2*x^4 - 4*b^2*(1 + a/(b*x^2))^(1/3)*x^4*Hypergeometric2F
1[1/3, 1/3, 4/3, -(a/(b*x^2))])/(12*a^2*x^4*(a + b*x^2)^(1/3))

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}}{\frac{1}{\sqrt [3]{b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x^2+a)^(1/3),x)

[Out]

int(1/x^5/(b*x^2+a)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(1/3)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222643, size = 196, normalized size = 1.42 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{\frac{1}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + a\right ) - 4 \, \sqrt{3} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} - a\right ) - 12 \, b^{2} x^{4} \arctan \left (\frac{2 \, \sqrt{3}{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (4 \, b x^{2} - 3 \, a\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{\frac{1}{3}}\right )}}{108 \, a^{\frac{7}{3}} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(1/3)*x^5),x, algorithm="fricas")

[Out]

-1/108*sqrt(3)*(2*sqrt(3)*b^2*x^4*log((b*x^2 + a)^(2/3)*a^(1/3) + (b*x^2 + a)^(1
/3)*a^(2/3) + a) - 4*sqrt(3)*b^2*x^4*log((b*x^2 + a)^(1/3)*a^(2/3) - a) - 12*b^2
*x^4*arctan(1/3*(2*sqrt(3)*(b*x^2 + a)^(1/3)*a^(2/3) + sqrt(3)*a)/a) - 3*sqrt(3)
*(4*b*x^2 - 3*a)*(b*x^2 + a)^(2/3)*a^(1/3))/(a^(7/3)*x^4)

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Sympy [A]  time = 5.55842, size = 41, normalized size = 0.3 \[ - \frac{\Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 \sqrt [3]{b} x^{\frac{14}{3}} \Gamma \left (\frac{10}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(b*x**2+a)**(1/3),x)

[Out]

-gamma(7/3)*hyper((1/3, 7/3), (10/3,), a*exp_polar(I*pi)/(b*x**2))/(2*b**(1/3)*x
**(14/3)*gamma(10/3))

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GIAC/XCAS [A]  time = 0.598851, size = 171, normalized size = 1.24 \[ \frac{1}{36} \, b^{2}{\left (\frac{4 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{7}{3}}} - \frac{2 \,{\rm ln}\left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{7}{3}}} + \frac{4 \,{\rm ln}\left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{7}{3}}} + \frac{3 \,{\left (4 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} - 7 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a\right )}}{a^{2} b^{2} x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(1/3)*x^5),x, algorithm="giac")

[Out]

1/36*b^2*(4*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3))/
a^(7/3) - 2*ln((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3)
+ 4*ln(abs((b*x^2 + a)^(1/3) - a^(1/3)))/a^(7/3) + 3*(4*(b*x^2 + a)^(5/3) - 7*(b
*x^2 + a)^(2/3)*a)/(a^2*b^2*x^4))